The Sudarshan-Glauber P representation is a suggested way of writing down the phase space distribution of a quantum system in the phase space formulation of quantum mechanics. The P representation is the quasiprobability distribution in which observables are expressed in normal order. In quantum optics, this representation, formally equivalent to several other representations, is sometimes preferred over such alternative representations to describe light in optical phase space, because typical optical observables, such as the particle number operator, are naturally expressed in normal order. It is named after George Sudarshan and Roy J. Glauber, who worked on the topic in 1963. Despite many useful applications in laser theory and coherence theory, the Glauber–Sudarshan P representation has the peculiarity that it is not always positive, and is not a bona-fide probability function. Definition We wish to construct a function with the property that the density matrix is diagonal in the basis of coherent states , i.e., We also wish to construct the function such that the expectation value of a normally ordered operator satisfies the optical equivalence theorem. This implies that the density matrix should be in anti-normal order so that we can express the density matrix as a power series Inserting the identity operator we see that and thus we formally assign More useful integral formulas for are necessary for any practical calculation. One method is to define the characteristic function and then take the Fourier transform Another useful integral formula for is Note that both of these integral formulas do not converge in any usual sense for "typical" systems . We may also use the matrix elements of in the Fock basis . The following formula shows that it is always possible to write the density matrix in this diagonal form without appealing to operator orderings using the inversion (given here for a single mode), where and are the amplitude and phase of . Though this is a full formal solution of this possibility, it requires infinitely many derivatives of Dirac delta functions, far beyond the reach of any ordinary tempered distribution theory. Discussion If the quantum system has a classical analog, e.g. a coherent state or thermal radiation, then is non-negative everywhere like an ordinary probability distribution. If, however, the quantum system has no classical analog, e.g. an incoherent Fock state or entangled system, then is negative somewhere or more singular than a Dirac delta function. (By a theorem